University of Utah
Teachers’ Circle
October 21, 2013
Questions in Probability
Instructions: Tonight we will explore three questions in probability. We will begin by investigating each
question using a hands-on activity. After we use the activities to predict answers to the questions, you will
decide which question you wish to pursue in more detail.
Question 1 If you give your students a matching test with n questions and all of your students just blindly
guess on the exam, what would you expect the class average to be?
Question 2 If you have a bowl of spaghetti with n noodles and you randomly tie the ends of all the
noodles together, how many spaghetti loops would you expect to have?
Question 3 Are there more obtuse triangles or acute triangles?
University of Utah
Teachers’ Circle
October 21, 2013
Activity 1:
In this activity we are investigating Question 1. Below is a matching test with 10 questions expect the
questions are missing. Randomly guess the correct answers and see how good you are at life.
Life’s Final Exam
1.
a. 1492
2.
b. Elephants
3.
c. Cell phones
4.
d. Goats
5.
e. Fireworks
6.
f. Spaghetti
7.
g. Matches
8.
h. Triangles
9.
i. Einstein
10.
Your score:
Compute the class average:
j. Fish
University of Utah
Teachers’ Circle
October 21, 2013
Activity 2
In this activity we investigate Question 2. Playing with actual spaghetti would be a mess. Instead you are
presented with 8 strings which simulates a bowl of spaghetti with 8 noodles. Randomly tie together the
ends of the loops. After you have finished tying the knot(s) answer the following questions.
1. How many loops were you left with?
2. Were any of the loops linked?
3. Two links are in the same link component if you cannot separate them without untying any knots.
4. What is the average length of your loops (measured in a new scientific unit - SNL - standard noodle
lengths)?
5. What was the average number of loops among everyone here tonight?
University of Utah
Teachers’ Circle
October 21, 2013
Activity 3
In this activity we investigate Question 3. You have been given 3 random integer points in R2 . Is the
triangle formed by these three points obtuse or acute.
1. Plot your 3 points.
2. Is your triangle obtuse or acute?
1
3. How many people tonight ended up with an acute triangle? How many people ended up with an
obtuse triangle?
1 If it is not clear just looking at the triangle, try and use a modification of the Pythagorean theorem to decide if your triangle
is obtuse or acute. To do this label the three sides of your triangle A, B,C with C being the longest side. Let a, b, c be the lengths
of A, B,C respectively. If c2 = a2 + b2 , it is a right triangle. If c2 > a2 + b2 , it is an obtuse triangle. If c2 < a2 + b2 , it is an acute
triangle.
University of Utah
Teachers’ Circle
October 21, 2013
Question 1
Below are some leading questions/hints to get you thinking about how to solve question 1.
1. Fill in the following table:
Number of questions
Expected value
1
2
3
4
2. With the data we collected from our matching exam and the above table, make a hypothesis about
what you expect the class average to be with n questions.
3. What is the probability that a student gets question 1 correct?
4. What is expected number of points a student earns on any one question?
5. Use part 4 to solve the problem.
Question 2
Below are some leading questions/hints to get you thinking about how to solve question 2.
1. Fill in the following table:
Number of noodles
Expected number of loops
1
2
3
4
2. If you have n noodles and and you choose an end and tie it at random with any of the remaining ends,
what is the probability you made a loop? If you didn’t make a loop, what is the question similar to
now? If you did make a loop what is the question similar to now?
3. Let Sn be the expected number of loops you make with n noodles. Use 2 to get a recursive
relationship between Sn and Sn−1 .
4. Use 3 to write a sum representing Sn .
Question 3
Below are some steps to help you organize your thoughts.
1. Let A, B,C be three random points.
2. Assume that the largest side length of the triangle 4ABC is side AB.
3. Assume that AB is horizontal (we can use some rotation for this to be true).
4. Draw two semi-circles centered around A and B each with diameter l(AB) (the length of the edge
AB).
5. Where must C be located? What is the area of this region?
6. Where must C be located for 4ABC to be obtuse? What is the area of this region?
7. Use parts 5 and 6 to calculate the probability that 4ABC is obtuse.
8. In this framework we assumed we had 3 randomly chosen points. What if we randomly choose our
angles? Would we get the same answer?